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Thread: Complex Numbers Question, Involves Trig

  1. #1
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    Complex Numbers Question, Involves Trig

    Hi, I'm looking for some help with complex numbers. I've attached a question below with the solution.

    I understand where the route 41 is coming from but I am getting a different angle.

    Part (b) I am also confused by.

    Help is much appreciated.
    Attached Thumbnails Attached Thumbnails Complex Numbers Question, Involves Trig-complex-numbers-q-jan-2014.jpg   Complex Numbers Question, Involves Trig-complex-numbers-ans-jan-2014.jpg  
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  2. #2
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    Re: Complex Numbers Question, Involves Trig

    Do you know that sin(a+ b)= sin(a)cos(b)+ cos(a)sin(b)?

    5cos(x)- 4 sin(x) cannot be "cos(a+ b)" for any a and b because "5" and "4" cannot be "sin(a)" and "cos(a)" for any a. sin(a) and cos(a) must be between -1 and 1 for all a and $\displaystyle sin^2(a)+ cos^2(a)= 1$. However, I note that $\displaystyle 5^2+ 4^2= 25+ 16= 41$ so that
    $\displaystyle 5 cos(x)- 4 sin(x)= \sqrt{41}\left(\frac{5}{\sqrt{41}}cos(x)- \frac{4}{\sqrt{41}}sin(x)\right)$.

    Now, what angle, a, has $\displaystyle sin(a)= \frac{5}{\sqrt{41}}$ and $\displaystyle cos(a)= -\frac{4}{\sqrt{41}}$?
    Last edited by HallsofIvy; Dec 29th 2014 at 10:52 AM.
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  3. #3
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    Re: Complex Numbers Question, Involves Trig

    Right, thanks for your help.

    For sin(a) I get 51.3 degrees and cos(a) - 38.7/321.3 degrees.

    How does this relate to the answers given in the solution?
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  4. #4
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    Re: Complex Numbers Question, Involves Trig

    No, this is the same angle a for both. Sine is positive in the first and second quadrants while cosine is negative in the second and third quadrants. In order that sine be positive and cosine negative, the angle must be in the second quadrant- 180- 51.3= 128.7 degrees.
    $\displaystyle sin(128.7)= 0.780= \frac{5}{\sqrt{41}}$ and $\displaystyle cos(128.7)= -0.63= -\frac{4}{\sqrt{41}}$

    Now, remember the point was that $\displaystyle 5cos(x)- 4sin(x)= \sqrt{41}(sin(a) cos(x)+ cos(a)sin(x))= \sqrt{41}sin(x+ a)$
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  5. #5
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    Re: Complex Numbers Question, Involves Trig

    Right I get that, I remember the quadrants now. So from 128.7 how are the final two x values calculated?

    Sorry but it's been a while since I've done this type of calculation.
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